Integrand size = 86, antiderivative size = 26 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{x+\left (\frac {2}{x}+x+\log \left (e^{4/5} x\right )\right ) \log ^2\left (x^2\right )} \]
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\[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=\int \frac {\exp \left (\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}\right ) \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx \\ & = \int \left (\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right )+\frac {4 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \left (10+4 x+5 x^2+5 x \log (x)\right ) \log \left (x^2\right )}{5 x^2}+\frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) (-1+x) (2+x) \log ^2\left (x^2\right )}{x^2}\right ) \, dx \\ & = \frac {4}{5} \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \left (10+4 x+5 x^2+5 x \log (x)\right ) \log \left (x^2\right )}{x^2} \, dx+\int \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \, dx+\int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) (-1+x) (2+x) \log ^2\left (x^2\right )}{x^2} \, dx \\ & = \frac {4}{5} \int \left (5 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )+\frac {10 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2}+\frac {4 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x}+\frac {5 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log (x) \log \left (x^2\right )}{x}\right ) \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \, dx+\int \left (\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )-\frac {2 \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x^2}+\frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x}\right ) \, dx \\ & = -\left (2 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x^2} \, dx\right )+\frac {16}{5} \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x} \, dx+4 \int \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right ) \, dx+4 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log (x) \log \left (x^2\right )}{x} \, dx+8 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2} \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \, dx+\int \exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right ) \, dx+\int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x} \, dx \\ & = -\left (2 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x^2} \, dx\right )+\frac {16}{5} \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x} \, dx+4 \int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \log \left (x^2\right ) \, dx+4 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log (x) \log \left (x^2\right )}{x} \, dx+8 \int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log \left (x^2\right )}{x^2} \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \, dx+\int e^{x+\left (\frac {4}{5}+\frac {2}{x}+x+\log (x)\right ) \log ^2\left (x^2\right )} \log ^2\left (x^2\right ) \, dx+\int \frac {\exp \left (x+\frac {\left (2+x^2\right ) \log ^2\left (x^2\right )}{x}+\log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )\right ) \log ^2\left (x^2\right )}{x} \, dx \\ \end{align*}
Time = 5.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{x+\left (\frac {4}{5}+\frac {2}{x}+x\right ) \log ^2\left (x^2\right )} x^{\log ^2\left (x^2\right )} \]
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Time = 1.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35
method | result | size |
parallelrisch | \({\mathrm e}^{\frac {x \ln \left (x^{2}\right )^{2} \ln \left (x \,{\mathrm e}^{\frac {4}{5}}\right )+\left (x^{2}+2\right ) \ln \left (x^{2}\right )^{2}+x^{2}}{x}}\) | \(35\) |
risch | \(x^{\frac {8 i \pi \,\operatorname {csgn}\left (i x \right )}{x}} x^{-\frac {8 i \pi \,\operatorname {csgn}\left (i x^{2}\right )}{x}} x^{4 i x \pi \,\operatorname {csgn}\left (i x \right )} x^{-4 i x \pi \,\operatorname {csgn}\left (i x^{2}\right )} x^{-\frac {16 i \pi \,\operatorname {csgn}\left (i x^{2}\right )}{5}} x^{\frac {16 i \pi \,\operatorname {csgn}\left (i x \right )}{5}} x^{-\frac {\pi ^{2}}{2}} x^{2 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )} x^{-\frac {3 \pi ^{2}}{2}} {\mathrm e}^{\frac {-5 x^{2} \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+20 x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-30 x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+20 x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-5 x^{2} \pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+16 x \,\pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-24 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+16 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-4 x \,\pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+80 i x \ln \left (x \right )^{2} \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-40 i x \ln \left (x \right )^{2} \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-40 i x \ln \left (x \right )^{2} \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-10 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6}+40 \pi ^{2} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{5}-60 \pi ^{2} \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )^{4}+40 \pi ^{2} \operatorname {csgn}\left (i x \right )^{3} \operatorname {csgn}\left (i x^{2}\right )^{3}-10 \pi ^{2} \operatorname {csgn}\left (i x \right )^{4} \operatorname {csgn}\left (i x^{2}\right )^{2}+80 x \ln \left (x \right )^{3}+80 x^{2} \ln \left (x \right )^{2}+64 x \ln \left (x \right )^{2}+160 \ln \left (x \right )^{2}+20 x^{2}}{20 x}}\) | \(537\) |
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Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{\left (\frac {5 \, x \log \left (x^{2}\right )^{3} + 2 \, {\left (5 \, x^{2} + 4 \, x + 10\right )} \log \left (x^{2}\right )^{2} + 10 \, x^{2}}{10 \, x}\right )} \]
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Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{\frac {x^{2} + x \left (\frac {\log {\left (x^{2} \right )}}{2} + \frac {4}{5}\right ) \log {\left (x^{2} \right )}^{2} + \left (x^{2} + 2\right ) \log {\left (x^{2} \right )}^{2}}{x}} \]
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Time = 0.40 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=e^{\left (4 \, x \log \left (x\right )^{2} + 4 \, \log \left (x\right )^{3} + \frac {16}{5} \, \log \left (x\right )^{2} + x + \frac {8 \, \log \left (x\right )^{2}}{x}\right )} \]
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\[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left ({\left (x^{2} + x - 2\right )} \log \left (x^{2}\right )^{2} + 4 \, x \log \left (x^{2}\right ) \log \left (x e^{\frac {4}{5}}\right ) + x^{2} + 4 \, {\left (x^{2} + 2\right )} \log \left (x^{2}\right )\right )} e^{\left (\frac {x \log \left (x^{2}\right )^{2} \log \left (x e^{\frac {4}{5}}\right ) + {\left (x^{2} + 2\right )} \log \left (x^{2}\right )^{2} + x^{2}}{x}\right )}}{x^{2}} \,d x } \]
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Time = 13.76 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {x^2+\left (2+x^2\right ) \log ^2\left (x^2\right )+x \log \left (e^{4/5} x\right ) \log ^2\left (x^2\right )}{x}} \left (x^2+\left (8+4 x^2\right ) \log \left (x^2\right )+4 x \log \left (e^{4/5} x\right ) \log \left (x^2\right )+\left (-2+x+x^2\right ) \log ^2\left (x^2\right )\right )}{x^2} \, dx=x^{{\ln \left (x^2\right )}^2}\,{\mathrm {e}}^{\frac {2\,{\ln \left (x^2\right )}^2}{x}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (x^2\right )}^2}{5}}\,{\mathrm {e}}^x\,{\mathrm {e}}^{x\,{\ln \left (x^2\right )}^2} \]
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